Abstract
For q> 1 we consider expansions in base q with digits set { 0 , 1 , q}. Let U q be the set of points which have a unique q-expansion. For k= 2 , 3 , … , ℵ let B k be the set of bases q> 1 for which there exists x having precisely k different q-expansions, and for q∈ B k let Uq(k) be the set of all such x’s which have exactly k different q-expansions. In this paper we show that Bℵ0=[2,∞)andBk=(qc,∞)for anyk≥2,where q c ≈ 2.32472 is the appropriate root of x 3 - 3 x 2 + 2 x- 1 = 0. Moreover, we show that for any integer k≥ 2 and any q∈ B k the Hausdorff dimensions of Uq(k) and U q are the same, i.e., dimHUq(k)=dimHUqfor anyk≥2.Finally, we conclude that the set of points having a continuum of q-expansions has full Hausdorff dimension.
| Original language | English |
|---|---|
| Pages (from-to) | 1605-1619 |
| Number of pages | 15 |
| Journal | Mathematische Zeitschrift |
| Volume | 291 |
| Issue number | 3-4 |
| DOIs | |
| Publication status | Published - 1 Apr 2019 |
Funding
Acknowledgements The second author was supported by NSFC no. 11701302 and K. C. Wong Magna Fund at Ningbo University. The third author was supported by NSFC no. 11401516 and Jiangsu Province Natural Science Foundation for the Youth no. BK20130433. The forth author was supported by NSFC nos. 11271137, 11571144, 11671147 and in part by Science and Technology Commission of Shanghai Municipality (no. 18dz2271000) Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Keywords
- Countable expansion
- Hausdorff dimension
- Multiple expansion
- Unique expansion